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Let $\mathcal{B}$ be a base for a topological space $X$. We denote by $\mathbf{PSh}_\mathcal{B}$ the category of presheaves on the base $\mathcal{B}$, and by $\mathbf{PSh}$ the category of presheaves on $X$.
Consider $i: \mathbf{PSh} \to \mathbf{PSh}_\mathcal{B}$ the inclusion functor. If $i$ were to admit a left ad... |
Is extension of a presheaf on a base to a presheaf on the whole space a left adjoint? |
I have $f(x, y)$ where $y$ is a function of $x$. So $f(x,y)=f(x, y(x))$. How do I differentiate the partial derivative
$$\frac{d}{dx}\left(\frac{\partial f}{\partial y}\right)=?$$ |
I have $f(x, y)$ where $y$ is a function of $x$. So $f(x,y)=f(x, y(x))$. How do I differentiate the partial derivative
$$\frac{d}{dx}\left(\frac{\partial f}{\partial y}\right)=?$$
$$\frac{d}{dx}\left(\frac{\partial f}{\partial x}\right)=?$$ |
I have $f(x, y)$ where $y$ is a function of $x$. So $f(x,y)=f(x, y(x))$. How do I differentiate the partial derivative
$$\frac{d}{dx}\left(\frac{\partial f}{\partial y}\right)=?$$
$$\frac{d}{dy}\left(\frac{\partial f}{\partial x}\right)=?$$ |
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1.$$
In the paper, they presented the vertices and find the maximum value.
These vertices are
$\left(\frac{n-1}{n... |
How to find vertices of intersection of two hyperplanes? |
I was looking at algebraic rules that certain functions follow. For example, exponential functions have the rule:
$e^a e^b = e^{a+b}$
And I was wondering if this is the only function that obeys this rule. So I set up the equation:
$z(x+y)=z(x)z(y)$
and tried taking partial derivatives on both sides. First w... |
Can you find a certain function that obeys a specific algebraic rule? |
I was solving a electrostatic problem which involved solving Laplace equation in two dimension. The solution is (in the form of Fourier sine series):
$$V(x,y)= \frac{4V_0}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1}exp(-(2n-1)\pi \frac xa)\sin ((2n-1)\pi \frac ya)$$
Now it asked to find the surface charger density ... |
I was looking at algebraic rules that certain functions follow. For example, exponential functions have the rule:
$e^a e^b = e^{a+b}$
And I was wondering if this is the only function that obeys this rule. So I set up the equation:
$z(x+y)=z(x)z(y)$
and tried taking partial derivatives on both sides. First w... |
From theoretical perspective, we define $X$ is a real-valued random variable from $(\Omega, \mathcal{A})$ to $(\Bbb{R},\mathcal{B}(\Bbb{R}))$ iff it is a measurable function wrt to $\mathcal{A}$ and $\mathcal{B}(\Bbb{R})$. To calculate the probability of events in $\mathcal{A}$, we add a measure $P$ on $(\Omega, \mathc... |
I was looking at algebraic rules that certain functions follow. For example, exponential functions have the rule:
$e^a e^b = e^{a+b}$
And I was wondering if this is the only function that obeys this rule. So I set up the equation:
$z(x+y)=z(x)z(y)$
and tried taking partial derivatives on both sides. First w... |
To find all $\displaystyle f\ \mathbb{N}\rightarrow \mathbb{N} \ $such that
\begin{gather}
\frac{f( x+y) \ +\ f( x)}{2x\ +\ f( y)} \ =\ \frac{2y\ +\ f( x)}{f( x\ +\ y) \ +\ f( y)} \notag\\
\notag\\
x\ =\ 1,\ y\ =\ 1 \notag\\
f( 2) \ +\ f( 1) \ =\ 2\ +\ f( 1) \notag\\
f( 2) \ =\ 2\\
x\ =\ 1,\ y\ =\ 2 \notag\\
... |
How to find all pairs $(a,b)$ of natural numbers that satisfy
$a^2-2b^2=-73696$ and $\gcd(a,b)=4$
The only thing I was able to do is to change the equation to
$a_1^2-2b_1^2=-4606$ with $\gcd(a_1,b_1)=1$
This gives that $2|a_1$ so the equation becomes $2a_1^2-b_1^2=-2303$
I can't find useful bound or facto... |
I'm taking a course in topology and we just defined the connected sum of connected surfaces (2-manifolds) as follows:
Given $S_1,S_2$ surfaces and $D_i\subseteq S_i$ open disks with $\varphi : \partial D_1 \rightarrow \partial D_2$ a homeomorphism the connected sum of $S_1,S_2$ is $S_1\#S_2=(S_1-D_1\cup S_2-D_2)/\va... |
Here is an example: https://www.desmos.com/calculator/h37wzmwdww
I have:
points at x1 = 0 and x2 = 90;
two parabolas with a1, b1, c1, a2, b2, c2.
I need the parabola a, b, c that is tangent to both parabolas by x1 and x2.
b=b1, c=c1, but how to get a?
The example:
a1 = 3.0133
b1 = -233.58
... |
Here is an example: https://www.desmos.com/calculator/h37wzmwdww
I have:
points at x1 = 0 and x2 = 90;
two parabolas with a1, b1, c1, a2, b2, c2.
I need the parabola a, b, c that is tangent to both parabolas at x1 and x2.
b=b1, c=c1, but how to get a?
The example:
a1 = 3.0133
b1 = -233.58
... |
I was trying to solve the question: how many positive integers $n \leq 100$ can be written as the sum of 4 or more consecutive integers.
Here was my approach:
Let $m$ represent the number of consecutive integers considered, and let $k$ be the starting integer.
For $m=4$:
$$ n = (k) + (k + 1) + (k + 2) + (k ... |
This is a question derived from the [17 camels problem][1]. I have been asked to find every integer $n$ satisfying
$$
n = 1+a+b+c,
$$
where $a, b, c$ are divisors of $n$ greater than one. By brute-forcing I got the answer $n\in\{12,18,20,24,42\}$:
\begin{align}
12 &= 1+2+3+6, \\
18 &= 1+2+6+9, \\
20 &= 1+4+5+10... |
To find all $\displaystyle f\ \mathbb{N}\rightarrow \mathbb{N} \ $such that
\begin{gather}
\frac{f( x+y) \ +\ f( x)}{2x\ +\ f( y)} \ =\ \frac{2y\ +\ f( x)}{f( x\ +\ y) \ +\ f( y)} \notag\\
\notag\\
x\ =\ 1,\ y\ =\ 1 \notag\\
f( 2) \ +\ f( 1) \ =\ 2\ +\ f( 1) \notag\\
f( 2) \ =\ 2\\
x\ =\ 1,\ y\ =\ 2 \notag\\
... |
Using a direct proof, how should I approach this question? |
Let $\mathcal{B}$ be a base for a topological space $X$. We denote by $\mathbf{PSh}_\mathcal{B}$ the category of presheaves on the base $\mathcal{B}$, and by $\mathbf{PSh}$ the category of presheaves on $X$.
Consider $i: \mathbf{PSh} \to \mathbf{PSh}_\mathcal{B}$ the inclusion functor. If $i$ were to admit a left ad... |
Consider the situation where we have two [cubic] bezier curves with the following properties:
- They share one common point (end of curve 1 = start of curve 2)
- They have the same direction at the point where they meet (a smooth join)
- We suspect that they are likely to be the result of having split a single ... |
This is a question derived from the [17 camels problem][1]. I have been asked to find every integer $n$ satisfying
$$
n = 1+a+b+c,
$$
where $a, b, c$ are distinct divisors of $n$ greater than one. By brute-forcing I got the answer $n\in\{12,18,20,24,42\}$:
\begin{align}
12 &= 1+2+3+6, \\
18 &= 1+2+6+9, \\
20 &=... |
> Does this mean that for any *particular* "ordinary" natural number that can be obtained by writing $S$ sufficiently many times in my formula, I can deduce $P(S(S(...S(0)))$ without appealing to the axiom of induction?
Yes, you can. To be exact, if you have
$$P(0)$$
and
$$\forall n (P(n) \to P(s(n)))$$
... |
In the case where the surface can be expressed in both spherical and Cartesian forms the result can be obtained by transforming the Cartesian integral into spherical coordinates. The theorem below gives a proof of this in terms of surface integrals - to obtain the result for surface areas choose an integrand function $... |
I'm trying to understand the following line in the proof to Proposition 5.3 in john M. Lee's 'Introduction to Smooth Manifolds':
"Because the projection $\pi_M : M \times N \rightarrow M$ satisfies $\pi_M \circ \gamma_f(x) = x$ for each $x \in U$ (where $U$) is an open subset of $M$, so the composition $d(\pi_M)_{(x... |
Why is this differential injective? Lee Smooth Manifolds Proposition 5.3 |
Is there a bound on how well you can (additive) approximate any natural number as the product of , say, first $10$ numbers - $\{1,2,3,\ldots,10\}$? By product, I mean just a single product, no additions allowed.
For example, using just the products of $\{1,2\}$, you can incur arbitrarily large errors. For example, a... |
How well can you approximate $\mathbb{Z}$ using product of first $t$ numbers? |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Suppose you have a right circular conical frustum with bottom base radius $r_1$ and top base radius $r_2$, and height $h$.
Now you take a (perspective) image of this frustum. The camera used to create the image has a known focal length, but its position and orientation are unknown.
Is it possible to determine fr... |
### Problem
Consider three positive numbers $a$, $b$ and $c$.
As the function $x \mapsto \sqrt{x}$ is concave, for any $(\alpha, \beta, \gamma)$ with $\alpha + \beta + \gamma = 1$ and $\alpha,\, \beta,\, \gamma \ge 0$:
\begin{equation}
\alpha \sqrt{a} + \beta \sqrt{b} + \gamma \sqrt{c} \le \sqrt{\alpha a + \... |
Move either the terms with $A$ or $B$ to the other side and square:
$$B^2\cos^2y=A^2\cos^2x-2AC\cos x+C^2\\B^2\sin^2y=A^2\sin^2x-2AD\sin x+D^2$$
Now add them together:
$$B^2=A^2+C^2+D^2-2AC\cos x-2AD\sin x$$
Now write $C=R\cos\phi$, $D=R\sin\phi$. So $R^2=C^2+D^2$.
$$B^2-A^2-C^2-D^2=-2A\sqrt{C^2+D^2}(\cos x\... |
**For the first part (The combinatorial form):** Suppose you have a group consisting of n boys and n girls. A team of n has to be formed out of them, and a captain designated of the team, which has to be a girl. On one hand, one can fix the number k of girls and n−k of boys on the team first, then choose the team in on... |
Under which minimal conditions on the function $f$ on $\mathbb{R}^+$ and the measure $\mu$ also on $\mathbb{R}^+$ we have:
$$ \int_A f d \mu \leq \mu(A) \int_A f d \lambda$$
for $\lambda$ Lebesgue measure and $A$ non empty measurable set? |
In the case where the surface can be expressed in both spherical and Cartesian forms the result can be obtained by transforming the Cartesian integral into spherical coordinates. The theorem below gives a proof of this in terms of surface integrals - to obtain the result for surface areas choose an integrand function $... |
**Context:**<br>
Hi, I started by calculating the chance of an event happening with a chance of 1 in x after n times. Next I wanted to flip this approach to be able to say how many n attempts are needed to have a given chance P(n) of an event happening.
**Examples:**<br>
1. With a chance of 1 in 4.2 billion km, h... |
Let us consider Fibonacci sequence modulo $41$. One can look up [the Pisano period](https://en.wikipedia.org/wiki/Pisano_period) of $41$, it is equal to $40$, but actually (the $0$-th and then) every $20$-th Fibonacci number is divisible by $41$. Here are Fibonacci numbers modulo $41$:
$$0, 1,1,2,3,5,8,13,21,34,14,7... |
How to find all pairs $(a,b)$ of natural numbers that satisfy
$a^2-2b^2=-73696$ and $\gcd(a,b)=4$
The only thing I was able to do is to change the equation to
$a_1^2-2b_1^2=-4606$ with $\gcd(a_1,b_1)=1$
This gives that $2|a_1$ so the equation becomes $2a_2^2-b_1^2=-2303$ ,$2a_2=a_1$
I can't find useful bou... |
Suppose that $\vec{x}\in\mathbb{R}^2$ is a vector and we want to find the minimum and the maximum of its components, using a two-layer neural network: $$\vec{y} = f_2(W_2f_1(W_1\vec{x}+b_1)+b_2), \ \ W_1,W_2 \in \mathbb{R^{2x2}},b_1,b_2\in \mathbb{R^2} \tag{1}\\ y_1 = \min(\vec{x}) = \min(x_1 , x_2) \\ y_2 = \max(\vec... |
I'm working on a potential flow problem. I have a box, centered at the origin (i.e. $V=[-x_b,x_b]\times[-y_b,y_b]\times[-z_b,z_b]$) that has inside of it a uniform distribution of source strength. We can assume this distribution has unit strength. The potential induced by the distribution of source strength in the box ... |
How to calculate the derivative of a Newtonian potential inside a box with uniform source distribution? |
I am currently reading a math paper in which the author writes :
$$\sum_{d\mid n}l\sigma_1(l)<<n^2\sigma_{-1}(n)$$
Where the function $\sigma_k$ is defined by $\sigma_k(n)=\sum_{d\mid n}d^k$ and does not give any detail for the proof. I have tried to do it by myself and show $\sum_{d\mid n}l\sigma_1(l)<<n\sigma_1(n)$... |
In the case where the surface can be expressed in both spherical and Cartesian forms the result can be obtained by transforming the Cartesian integral into spherical coordinates. The theorem below gives a proof of this in terms of surface integrals - to obtain the result for surface areas choose an integrand function $... |
I tried to read as many notes and answers possible before asking this, but I find myself in need of some answer from you guys.
So my confusion is about multivalued functions VS set valued functions VS vector functions.
To say, an example of multivalued function is the cubic root, in $\mathbb{C}$ of some complex ... |
It is clear if $a \geq 1$ or $b \geq 1$, but how can I show it when $0< a,b <1$?
I want hints.
|
How to prove $a^{b}+b^{a} \geq 1$ for all $a, b \geq 0$? |
Given a continuous stochastic process $x_t$ defined by the following Langevin equation
\begin{equation}
d x_t = dB_t +F(x_t)dt
\end{equation}
where $dB_t$ is a Wiener increment and $F(x_t)$ is a general force such that $F(x_t)=-V'(x_t)/2$. It can be shown that this process, in the steady state, is sampled by ... |
I have found this tough combinatorial geometry question in some Soviet math competition circa $1972$. It looks very simple (and rather natural), yet I cannot find the proof for quite some time now.
*$2n$ points are given on the plane. Prove that there exist $n$ non-overlapping circles, each containing exactly two gi... |
I am a retired philosopher, familiar with some of the philosophical problems about probability (e.g. Hume's problem of induction) but at a loss in calculating probabilities. I recently came across the following problem - problem for me, that is, not site users.
The number of pupils in a school in the years 2020, 20... |
Preface
=======
Consider a plane curve defined by parametric equations (for $t_1\le t \le t_2$):
$$x=x\left(t\right)$$
$$y=y\left(t\right)$$
In addition, there is a scalar function $f\left(x,y\right)$ defined for every value of $x$ and $y$.
Giving an example to illustrate (in this example, $t$ is the cent... |
How to parametrically express a parametric plane curve of a 3D plot as an axis in different 2D plot? |
I expected that this question was a duplicate, but I couldn't locate another instance of it.
> But is there such a thing as a "straight line" which is not distance minimal that can be cast between any two points such that the initial heading is the same as the final heading always? ... What would this line be called... |
I learn dg categories from Toën's lecture notes [Lectures on dg-categories][1], where the tensor product $T\otimes T'$ of two dg-cateogries is defined to be such that it has objects of pairs $(x,x')$ and mapping spaces as $$(T\otimes T')\big((x,x'),(y,y')\big)=T(x,y)\otimes T'(x',y')$$
This definition makes the catego... |
What's the Lurie's tensor product of dg categories? |
The question:
In a basket there are $20$ black balls, $15$ white balls and $18$ red balls. All balls with the same color are identical to each other. In how many ways it is possible to take out balls from the basket (when order doesn't matter), such that will take out at least one black ball, at least one white ball... |
In how many ways it is possible to take out balls from the basket , such that will take out at least one from each color? |
I am fairly new to proof writing so I would appreciate if you could help me out!
This is what I know: If $(X*, Y*)$ is a saddle point for a zero sum game with matrix A (which is an n x m matrix),
$\Rightarrow E(X,Y*) \leq E(X*,Y*) \leq E(X*,Y) \forall X \in S_n, Y \in S_m \\$
Where $E(X,Y)$ denotes the e... |
In ridge regression, we can estimate $\hat y$=$X(X^TX+\lambda I)^{-1}y$,where $X$ is covariate matrix with n rows and p column. And my teacher says that we can use SVD to rewrite this formula as:$\hat y$=$(XX^T+\lambda I)^{-1}XX^Ty$.
I have no idea about the last formula. Could you give me more details? |
I am fairly new to proof writing so I would appreciate if you could help me out!
This is what I know: If $(X*, Y*)$ is a saddle point for a zero sum game with
matrix A (which is an n x m matrix),
$\Rightarrow E(X,Y*) \leq E(X*,Y*) \leq E(X*,Y) \forall X \in S_n, Y \in S_m \\$
Where $E(X,Y)$ denotes t... |
i am trying to find the coefficients to a general solution for non-linear, autonomous second order ordinary differential equation, i have ended up with a general solution that has 4 integration constants. does this mean that i need 4 initial conditions? and if so which? i know that i can already use the initial conditi... |
how many initial conditions are needed for a specific solution to a non-linear, autonomous second order differential equation? |
I am currently reading a math paper in which the author writes :
$$\sum_{l\mid n}l\sigma_1(l)<<n^2\sigma_{-1}(n)$$
Where the function $\sigma_k$ is defined by $\sigma_k(n)=\sum_{d\mid n}d^k$ and does not give any detail for the proof. I have tried to do it by myself and show $\sum_{l\mid n}l\sigma_1(l)<<n\sigma_1(n)$... |
I am currently reading a math paper in which the author writes :
$$\sum_{l\mid n}l\sigma_1(l)\ll n^2\sigma_{-1}(n)$$
Where the function $\sigma_k$ is defined by $\sigma_k(n)=\sum_{d\mid n}d^k$ and does not give any detail for the proof. I have tried to do it by myself and show $\sum_{l\mid n}l\sigma_1(l)\ll n\sigma_1... |
Let $E$ be a vector bundle over a topological space $M$. I want to show that $\pi:E \to M$ is a homotopy equivalence.
To show this, I use the zero section $\zeta:M \to E$. Then $\pi \circ \zeta = Id_M$ and $\zeta \circ \pi$ is the zero map in the vector space $E_p$ for $p\in M$.
So the natural homotopy map is $... |
How to show that the map $E \times I \to E: (v,t) \to t\cdot v$ is continuous on a vector bundle $E$? |
I'm wondering about some possibility to modify Taylor series, due to it's pragmatic implication on operator of indefinite integration and summation.
Let's expand Taylor series to the form $f(x)=\sum_{n=-\infty}^{\infty}\frac{f^{(n)}(0)x^n}{n!}$.
Because for negative integers $n$, $\frac{1}{n!} =0$, series at th... |
I am trying to use the least squares regression to fit a curve to a table of values representing a sine wave. Similar to the question:
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
Except I want to fit a more general formula of: $a * sin(b*x + c) + d$
So given a function ... |
Let $A$ and $B$ be finite sets of points in $\mathbb{R}^2$. Let $C$ be the set of all images of $A$ under isometries of the plane (translations, rotations, and reflections, but no scaling). We say that $C$ [*shatters*](https://en.wikipedia.org/wiki/Shattered_set) $B$ (or that $A$ shatters $B$ up to isometry) if
$$\... |
What is the smallest point set that shatters a set of five points up to isometry? |
>Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$.
It's quite clear that $\mathbb{Q}(\sqrt[d]a)$ must have $x^d-a$ as its minimal polynomial in order to have degree $d$. And also... |
Imagine that you have a fraction, where both the numerator and denominator take values in the set $\{1,2,\dots, n\}$. Let us assume that the fraction is smaller then and or eqal to 1. The question is how many unique values can you get with every single combination in relation to $n$ and is there a function to describe ... |
Consider the following;
$$
J = L^{\text{T}}KR
$$
where
$$
L \in \mathbb{R}^{p}, ~~ K \in \mathbb{R}^{p \times m}, ~~ r \in \mathbb{R}^{m}
$$
I am attempting to find $ \frac{dJ}{dK}$. This is what I have done so far;
\begin{align}
J &= \begin{bmatrix}
\ell_1 & \ell_2 & \dotsm & \ell_p
\e... |
I am trying to use the least squares regression to fit a curve to a table of values representing a sine wave. Similar to the question:
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
Except I want to fit a more general formula of: $a * \sin(b*x + c) + d$
So given a function... |
Let $p$ be a prime, $d$ a proper divisor of $p-1$, and $\sigma\in S_{p-1}$ of the form (everything is modulo $p$):
$$\sigma=(1,x,\dots,x^{d-1})(i_2,i_2x,\dots,i_2x^{d-1})\dots(i_k,i_kx,\dots,i_kx^{d-1})\tag1$$
(hence $|\sigma|=d$) where $kd=p-1$ and, $\forall i,j=1,\dots,p-1$ such that $i+j\not\equiv_p0$:
$$\sigma(i... |
Let $k$ denote the field of real numbers, or the field of complex numbers. Given a vector space $V$ over $k$, it is always possible define a metric on $V$ (say the [discrete metric][1]), or a norm on $V$ ([by choosing a Hamel basis][2]). However, if the norm you define has no relation to the metric you define, then the... |
> Let $K_{1}$ and $K_{2}$ be two isomorphic fields. Prove that an isomorphism $f \colon K_{1} \xrightarrow{\sim} K_{2}$ can be extended to an isomorphism $\sigma \colon \overline{K_{1}} \xrightarrow{\sim} \overline{K_{2}}$ between their respective closures.
I intended to do this using Zorn's Lemma (akin to the proo... |
>Suppose functions $y$ and $f$ are non-negative and continuous on the interval $<a,b>$ (where $<$ can be $( $ or $[ $ and $>$ can be $] $ or $) $), and let $\lambda > 0$. If the inequality
$
y(x) \leq f(x) + \lambda \Big| \int_{x_0}^x y(s) \, ds \Big|, \quad x, x_0 \in (a,b)
$
holds, then
$
y(x) \leq f(x) + \lam... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
Assuming you know that the eccentricity of hyperbola is
$$e=\sqrt{1+\frac{b^2}{a^2}}$$
and that the coordinate of foci is
$$(\pm c,0) or, (\pm ae,0)$$
this means the distance between the focus and center is
$$c = ae$$
$$=a\sqrt{1+\frac{b^2}{a^2}}$$
$$=\sqrt{a^2(1+\frac{b^2}{a^2})}$$
$$=\sqrt{a^2+a^2... |
This is the definition of a restriction of a vector bundle given in John Lee's Introduction to Smooth Manifolds.
Let $E,M$ be smooth manifolds with or without boundary and $\pi:E \to M$ a rank-$k$ smooth vector bundle and $S$ be an embedded submanifold with or without boundary of $M$. Then we have the restriction of... |
The order of a Galois group, $G$, of a polynomial of the type $X^n-a$ over $\mathbb{Q}$, with $a, n \in \mathbb{N}$ is $|G|=n\cdot\varphi(n)$ when $n$ and $\varphi(n)$ are coprime (although I'm not sure about this condition). For instance, if $f(X)=X^5-2$, the Galois group has $5 \cdot\varphi(5)=5 \cdot 4 = 20$ element... |
Too long for comment and also don’t have enough reputation
I had hoped that you would have started the calculations, especially as $f(\xi,y)$ occurs on the right side of $$\dfrac{\partial^{n+1} f(\xi,y)}{\partial^{n+1} x}$$ which makes the term
$$
\dfrac{\partial^{n+1} f(\xi,y)}{\partial^{n+1} x}\prod_{i=0}^n(x... |
I am trying to represent the circle of radius 2 in polar coordinates.
$$\gamma(t)=(2\cos(t),2\sin(t))$$
where $t\in [0,2\pi]$. The derivative would be
$$\gamma'(t)=-2\sin(t)\partial_x +2\cos(t)\partial_y.$$
Now if we convert the tangent basis vectors to polar coordinates,
$$\partial_x = \cos(t)\partial... |
> Let $K_{1}$ and $K_{2}$ be two isomorphic fields. Prove that an isomorphism $f \colon K_{1} \xrightarrow{\sim} K_{2}$ can be extended to an isomorphism $\sigma \colon \overline{K_{1}} \xrightarrow{\sim} \overline{K_{2}}$ between their respective closures.
I intended to do this using Zorn's Lemma (akin to the proo... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
> Let $K_{1}$ and $K_{2}$ be two isomorphic fields. Prove that an isomorphism $f \colon K_{1} \xrightarrow{\sim} K_{2}$ can be extended to an isomorphism $\sigma \colon \overline{K_{1}} \xrightarrow{\sim} \overline{K_{2}}$ between their respective closures.
I intended to do this using Zorn's Lemma (akin to the proo... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
Let's say I have two functions $f$ and $g$, both analytic on the same domain (open and connected set) $D$, and suppose also I was able to prove that $f = 0$ on the entire domain.
**Question**. Is it legitimate to then claim that
$$
\frac{f(z)}{g(z)}\equiv 0
$$ identically on that domain?
In particular, $g(z)$ ... |
If an analytic function is identically zero in a domain, can it be always divided by another analytic function in that domain? |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
> Does this mean that for any *particular* "ordinary" natural number that can be obtained by writing $S$ sufficiently many times in my formula, I can deduce $P(S(S(...S(0)))$ without appealing to the axiom of induction?
Yes, you can. To be exact, if you have
$$P(0)$$
and
$$\forall n (P(n) \to P(s(n)))$$
... |
When: $ \int_A f d \mu \leq \mu(A) \int_A f d \lambda$? |
Let $\mathcal{L}$ be the set of Lebesgue measurable subsets of $(0,1]$ such that $\lambda(A+x)=\lambda(A)$ where $A+x:=\{y\in (0,1]: y-x\in A\}$. I try to show that $\mathcal{L}$ is a $\lambda$-system.
I check definition:
(1) Clearly, $(0,1]\in \mathcal{L}$.
But I am stuck on the another ones
(2) for $E_... |
Denote $(a,b,c):=(x_b,y_b,z_b)$, $(x,y,z):=(x_0,y_0,z_0)$, and $I:=\left.\dfrac{\partial^2\phi}{\partial x^2}\right\rvert_{(0,0,0)}$.
Integrating by parts makes quick work of the $x$-integral,
$$-3 \int_0^a \frac{x^2}{\left(x^2+y^2+z^2\right)^{5/2}} \, dx = \frac a{\left(a^2+y^2+z^2\right)^{3/2}} - \int_0^a \frac... |
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